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Equations for the Evaluation of Thermodynamic Quantities for Multicomponent Systems Sebastian G. Canagaratna*,† and M. Maheswaran*,‡ † ‡
Department of Chemistry and Biochemistry, Ohio Northern University, Ada, Ohio 45810, United States Department of Mathematics, University of Wisconsin—Marathon County, Wausau, Wisconsin 54401, United States ABSTRACT: Simple mathematical transformations on the Gibbs excess function GEm permit the evaluation of many important thermodynamic quantities. It is therefore of considerable interest to be able to evaluate this excess function from experimental data. We extend the methods already developed for excess functions based on mole-fractions to derive appropriate partial differential equations which relate any pair of GEm, the logarithm of an activity coefficient, or the osmotic coefficient φ. The derivation is general and does not depend on the experimental data conforming to any particular model. We illustrate the applications of these equations to experimental data from the literature. Equations and applications for other partial molar quantities are also discussed.
’ INTRODUCTION For binary systems, equations are available that permit the direct passage from one partial molar quantity to another or from a mean molar quantity to a partial molar quantity or vice versa. These equations are very general and do not assume any particular functional form for the quantities. Their derivation depends only on the properties of extensive functions and the GibbsDuhem equation and does not depend on any particular model. Surprisingly, no general equation has been derived for the direct passage from the osmotic coefficient φ to the Gibbs excess function on the molality scale, GEm. The Gibbs excess function is important and convenient because many thermodynamic properties may be derived from it by simple mathematical transformations. As part of an approach to deal with multicomponent electrolyte solutions, Pitzer1,2 developed an analysis that starts off from an assumed expression for GEm and derives equations for other quantities in terms of the coefficients (ion-interaction parameters) that appear in the expression for GEm. The equations are written in terms of ionic compositions, not molecular compositions. This method can in principle be used also for nonelectrolytes. Other models such as those due to Robinson and Stokes3 or the Lietzke and Stoughton model4 have been used, but the main interest in their models is to predict behavior of multicomponent systems from the properties of binary systems rather than in the evaluation of partial molar quantities. A disadvantage of approaches using model expressions is that when the experimental quantity does not strictly conform to the requirements of the model some modification of equations of the model may be required to fit experimental data. Instead of having to make adjustments to a specific model and revise its equations to conform to experimental data, it would be preferable to use an equation whose derivation is not constrained by any particular model but is easily adapted to the requirements of the experimental data. It is useful to recall equations that are available at present. McKay5 dealt with ternary systems, and the method he developed was directed to isopiestic measurements. Since it required values at r 2011 American Chemical Society
constant activity of solvent, some interpolations had to be carried out. This method does not appear to be used currently. The method developed by Darken6 for analyzing ternary systems has been used especially in alloy systems. He suggested extensions to multicomponent systems but did not give any explicit equations to analyze general multicomponent systems. Both McKay and Darken used cross-differentiation relations for their derivations. Neither treatment gave a unified treatment for any general partial molar property, and the resulting equations do not have the same form as those for binary systems. On the other hand, the differential equations approach taken by Canagaratna and Maheswaran7,8 derives equations for any general partial molar quantity, activity coefficients, as well as excess and mean molar quantities based on the mole fraction and molality scales. The derivation is straightforward and yields equations for multicomponent systems that are formally identical to those for binary systems. Darken’s method of keeping the ratios of solute compositions constant (effectively using pseudobinaries) arises naturally as the characteristics of a partial differential equation. Our goal in this paper is to develop equations whose derivations are independent of any model, so that it is possible to determine all partial molar quantities after a sufficiently precise fit has been obtained for the available experimental data. This is achieved by applying the partial differential equations approach to obtain equations for the excess quantity GEm based on molalities, as well as for the osmotic coefficient φ and activity coefficients γi. We illustrate the application of these equations by analyzing a few examples from the literature. Also, to illustrate how experimental data for any general quantity may be analyzed to obtain the corresponding partial molar quantities, we consider the example of molar volume.
’ RELATIONSHIPS AMONG φ, GEm, AND ln γi A common way of investigating a multicomponent system is to measure the osmotic coefficient φ. It is therefore of interest to Received: October 24, 2010 Accepted: April 23, 2011 Published: May 05, 2011 2761
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establish equations to go from φ to GEm and from φ to ln γi, where γi is the (mean) activity coefficient of solute species i. We derive equations that enable us to obtain analytical expressions for the excess free energy and the activity coefficient when a least-squares fit has been made for φ as a function of composition. Thermodynamic Preliminaries. For the discussion in this section we take the set of molalities of the constituents {m2, m3, ..., mN} to be the independent variables, where N is the total number of components. The solvent is denoted by subscript 1 and solutes are denoted by subscripts 2, 3, ..., N. Let W1, a1 and M1 be, respectively, the mass, the relative activity and molar mass of the solvent. Also, νi is the number of ions produced from one molecule of species i . The quantities GEm and φ are defined by Em
G ¼ W1 RT
N
∑ νi mi ð1 φ þ ln γi Þ N
∑ νi m i i¼2
GEm ¼ W1 RT
ð2Þ
N
∑ i¼2
d ln a1 ¼ M1 from which we obtain d ln a1 þ M1
N
N
∑ mi d ln ai i¼2 !
∑ mi ln ai i¼2
¼ M1
N
∑ ln ai dmi i¼2
ð4Þ
ð5Þ
For i,j = 2, 3, ..., N and i 6¼ j, eqs 5 and 3 yield the crossdifferentiation relations
νi
Dln aj Dln ai ¼ Dmj Dmi
ð6Þ
Dln γj Dln γi ¼ νj Dmj Dmi
ð7Þ
Relationship between GEm and ln γi. Combining eqs 1, 2,
and 4, we obtain the equation dðGEm =W1 RTÞ ¼
N
∑ νi ln γi dmi i¼2
ð8Þ
whence DðGEm =W1 RTÞ ¼ νi ln γi Dmi
ð9Þ
This equation is for neutral electrolyte species i and enables us to determine ln γi from a knowledge of GEm/W1RT. It
ð10Þ
mi
DðGEm =W1 RTÞ GEm ¼ Dmi W1 RT
N
∑ νi mi ð1 φÞ i¼2
ð11Þ equaalong as an eq 11
dmj dmi dm ¼ ¼ ::: ¼ m mi mj
ð3Þ
Qi is independent of composition and depends only on the number of cations and anions produced from a single neutral molecule. For constant T and p, the GibbsDuhem equation in terms of molalities reads
DðGEm =W1 RTÞ Dmi
N
According to the theory of linear partial differential tions, there are special paths called characteristics which a partial differential equation may be treated ordinary differential equation. The characteristics for are given by
A nonelectrolyte species j may be included in the treatment by putting νj = 1. Let γi be expressed on the molality scale. It is related to the relative activity ai by the relation ai ¼ ðQi mi γi Þνi
N
∑ νi mi ð1 φÞ þ i∑¼ 2 mi i¼2
This may be rearranged to read
ð1Þ
i¼2
ln a1 ¼ M1 φ
must be distinguished from Pitzer’s eq (14.23)9 which is for an ion. When ln γi is known, changes in GEm/W1RT may be obtained from eq 9 by integration with respect to mi, while all of the other mj are held constant. Differential Equation Connecting GEm and O. The following partial differential equation for GEm/W1RT is obtained by substituting for ln γi from eq 9 in eq 1:
¼
dðGEm =W1 RTÞ ðGEm =W1 RTÞ
N
∑
i¼2
ð12Þ
νi mi ð1 φÞ
Here, m = m2 þ m 3 þ ... þ m N is the total molality. According to eq 12, the ratio mi/mj of the molalities of any two solutes is constant along a characteristic. Consequently, it is convenient to change the variables from the set {m2, ..., mN} to the set {mi, m 2/mi, m3/mi, ..., mi1 /mi, miþ1/mi, ..., mN/mi}, where we choose any one of the molalities, mi, and the ratios of the other molalities to the selected molality. Then, when we move along a characteristic, only mi changes and the remaining variables in this set remain constant. In some instances, it may be preferable to use the set {m, m 2/m, m3/m, ..., mN/m}, where, instead of mi, we select the total molality m together with the ratios m i/m as the variables. However, in this case only N 2 of these ratios are independent. We use the expression differentiation along a characteristic to mean differentiation with respect to the selected variable in one of the above set of variables, while the ratios of the molalities are kept constant. We use the subscript “ch” to denote the differential coefficient along a characteristic. We note that when the ionic strength I is used, we have dI/I = dm/m and the ratio y i = Ii/I representing the ionic strength fraction is constant along a characteristic. Along a characteristic, we may rewrite eq 12 in the form " # N ðGEm =W1 RTÞ 1 d νj mj ð1 φÞd ð13Þ ¼ mi m i j¼2
∑
Note that in the above equation we may use the molality mi of any solute or replace it by m or I, whenever appropriate. To integrate eq 13 along a characteristic, we need to write all of the functions in terms of mi and the ratios mj/mi and regard the ratios as constants during the integration. Also, we note that (GEm/W1RT)/mi tends to 0 as the solution approaches infinite 2762
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0
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1
N
∑
νj m j Z B C C Bj ¼ 2 ðGEm =W1 RTÞ C dmi ¼ B ð1 φÞ C B 2 mi A 0 @ mi
ð14Þ
N
¼
∑ νjmjZ j¼2 mi
0
φ1 dmi mi
ð15Þ
Here we have moved ∑ νjmj/mi outside the integral sign because it is a constant along a characteristic given by mj/mi equal to a constant for all j. We finally obtain Z N φ1 GEm =W1 RT ¼ νj mj ð16Þ dmi mi 0 j¼2
∑
The integral on the right-hand side of eq 16 tends to zero as mi tends to zero even (as for electrolytes) if φ 1 tends to m1/2. This can be shown by carrying out the integration with respect to m1/2 as the variable of integration. The equation is identical in form to the equation for a binary solution, except that the integration in this case is carried out along a characteristic. After a functional form for φ has been obtained from a leastsquares fit, eq 16 enables us to derive an equation for GEm/ W1RT. Equation 9 can then be applied to obtain expressions for the activity coefficients. We illustrate this in the analysis of Robinson and Stokes’ data in the section on nonelectrolyte solutions. Differential Equation for ln γi. In some instances, it is useful to have a direct route from φ to ln γi. We combine eq 2 with eqs 4 and 7 to obtain the partial differential equation N
N
∑ mj j¼2
Dln γi ¼ φ1þ Dmj
∑ νj mj Dφ j¼2 νi
ð17Þ
Dmi
The characteristics for this differential equation are given by dmj dmi dm ¼ ¼ ¼ ::: ¼ m mi mj
d ln γi ðφ 1Þ þ ðDφ=Dmi Þ
N
∑ ν j mj
j¼2
! =νi
ð18Þ We integrate the above equation along characteristics starting from the state of infinite dilution, where ln γi = 0. We obtain N
∑
νj mjZ Z φ1 Dφ j¼2 ln γi ¼ dmi þ dmi mi νi mi 0 Dmi 0
ð19Þ
Along a characteristic, the integrands must be written as functions of the integration variable mi and the ratios mj/mi, which are treated as constants. It is important to note that the partial derivative of φ at the end of eq 19 is the regular partial derivative with respect to mi treating m2, m3, ..., mN as the variables. As stated earlier, the integration variable mi can be selected arbitrarily or may be replaced by the total molality m or the ionic strength I. Once information about φ as a function of composition is
known, eq 19 enables us to get an analytical expression for ln γi as a function of composition. The reverse process, which is the calculation of φ from experimental information of ln γi, may also be performed by using eq 17. However, only one partial derivative of φ is involved, and the integration has to be carried out along a line where only mi varies, while all of the other molalities are kept constant. We obtain the equation ! ! Z N N D ln γi νj m j φ ¼ νi mj þ 1 dmi þ constant Dmj mi ¼ 0 j ¼ 2 j¼2
∑
∑
ð20Þ To evaluate the constant of integration, we need to know the osmotic coefficient when mi = 0.
’ APPLICATION OF THE EQUATIONS We now illustrate the application of the above equations with experimental data from the literature. All curve fitting was carried out using the R package.10 This package has diagnostics which facilitate recognition of outliers as well as a test of normality of the distribution of the data points. A point worth noting about the least-squares fit using the R package is that when the expression used for the fit results in the overdetermination of the coefficients the program flags the terms which cause the problem. This overdetermination of the coefficients arises from linear relationships between terms; the exact form of the linear relationship can be obtained from R. In the theoretical development one uses all possible terms, but this creates problems when such a symmetrical expression is used for a least-squares fit. Thus for a binary system with variables x1 and x2, if an experimental quantity y is fitted to an expression of the form Ax1 þ Bx2 þ C, all three coefficients cannot be determined uniquely by a least-squares fit because of the relation x1 þ x2 = 1 between two of the terms. The only way of determining A,B, and C uniquely is when one of them, say C, is known independently of the fit (e.g., by determining y when x1 = 1 and x2 = 0). Then, C can be held constant during the fit. In a similar way, we cannot include all three terms x1x2, x12x2, and x1x22 in an expression for a least-squares fit because of the relationship x1x2 = x12x2 þ x1x22. The R package also reports the statistical significance of the coefficients that are determined. When any coefficients were not statistically significant, the least-squares fit was repeated after the corresponding terms had been removed. The dropping of some terms results in the expression for the fit not being symmetrical with respect to the variables representing the composition. This is evident in the illustrative examples we consider. Nonelectrolyte Solutions. We analyze the results of Robinson and Stokes11 on aqueous solutions of sucrose and mannitol. Robinson and Stokes developed a treatment that started from a cross-differentiation relation but assumed that the cross-differential could be represented as a sum of two functions, each depending only on one variable. Our analysis is direct and does not make any assumptions whatsoever. We pooled the data for the osmotic coefficient of the two binaries and the ternary system and carried out a linear least-squares analysis. The expression used to fit the data, after terms reported by R as not being statistically significant are dropped, is φ 1 ¼ A2 m2 þ A3 m3 þ A23 m2 m3 þ A2sq m22 þ A2sq3 m22 m3 þ A2cub m32 þ A2cub3 m32 m3 þ A2q m42 þ A3q m43 2763
ð21Þ
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Table 1. Sucrose-Mannitol: Coefficients of Fit for the Osmotic Coefficient O, eq 21a coeff.
estimate 6.789 10
A2
std. error
2
4
9.1 10
75
Pr(>|t|)
coeff.
16
